Whakaoti i te {l}{2y-3x=-27}{5y+3x=6}

$$\left. \begin{array} { l } { 2 y - 3 x = - 27 } \\ { 5 y + 3 x = 6 } \end{array} \right.$$

Whakaoti mō y, x

$$x=7$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$2y-3x=-27,5y+3x=6$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te y mā te wehe i te y i te taha mauī o te tohu ōrite.

$$2y-3x=-27$$

Me tāpiri 3x ki ngā taha e rua o te whārite.

$$2y=3x-27$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{1}{2}\left(3x-27\right)$$

Whakareatia \frac{1}{2} ki te -27+3x.

$$y=\frac{3}{2}x-\frac{27}{2}$$

Whakakapia te \frac{-27+3x}{2} mō te y ki tērā atu whārite, 5y+3x=6.

$$5\left(\frac{3}{2}x-\frac{27}{2}\right)+3x=6$$

Whakareatia 5 ki te \frac{-27+3x}{2}.

$$\frac{15}{2}x-\frac{135}{2}+3x=6$$

Tāpiri \frac{15x}{2} ki te 3x.

$$\frac{21}{2}x-\frac{135}{2}=6$$

Me tāpiri \frac{135}{2} ki ngā taha e rua o te whārite.

$$\frac{21}{2}x=\frac{147}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{21}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=7$$

Whakaurua te 7 mō x ki y=\frac{3}{2}x-\frac{27}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{3}{2}\times 7-\frac{27}{2}$$

Whakareatia \frac{3}{2} ki te 7.

$$y=\frac{21-27}{2}$$

Tāpiri -\frac{27}{2} ki te \frac{21}{2} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=-3$$

Kua oti te pūnaha te whakatau.

$$y=-3,x=7$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$2y-3x=-27,5y+3x=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\5&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-27\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\5&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\5&3\end{matrix}\right))\left(\begin{matrix}2&-3\\5&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&3\end{matrix}\right))\left(\begin{matrix}-27\\6\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&3\end{matrix}\right))\left(\begin{matrix}-27\\6\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&3\end{matrix}\right))\left(\begin{matrix}-27\\6\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-\left(-3\times 5\right)}&-\frac{-3}{2\times 3-\left(-3\times 5\right)}\\-\frac{5}{2\times 3-\left(-3\times 5\right)}&\frac{2}{2\times 3-\left(-3\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-27\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{1}{7}\\-\frac{5}{21}&\frac{2}{21}\end{matrix}\right)\left(\begin{matrix}-27\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\left(-27\right)+\frac{1}{7}\times 6\\-\frac{5}{21}\left(-27\right)+\frac{2}{21}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-3\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-3,x=7$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$2y-3x=-27,5y+3x=6$$

Kia ōrite ai a 2y me 5y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$5\times 2y+5\left(-3\right)x=5\left(-27\right),2\times 5y+2\times 3x=2\times 6$$

Whakarūnātia.

$$10y-15x=-135,10y+6x=12$$

Me tango 10y+6x=12 mai i 10y-15x=-135 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10y-10y-15x-6x=-135-12$$

Tāpiri 10y ki te -10y. Ka whakakore atu ngā kupu 10y me -10y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-15x-6x=-135-12$$

Tāpiri -15x ki te -6x.

$$-21x=-135-12$$

Tāpiri -135 ki te -12.

$$-21x=-147$$

Whakawehea ngā taha e rua ki te -21.

$$x=7$$

Whakaurua te 7 mō x ki 5y+3x=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$5y+3\times 7=6$$

Whakareatia 3 ki te 7.

$$5y+21=6$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$5y=-15$$

Whakawehea ngā taha e rua ki te 5.

$$y=-3$$

Kua oti te pūnaha te whakatau.

$$y=-3,x=7$$